energies Article Statistics to Detect Low-Intensity Anomalies in PV Systems Silvano Vergura * ID and Mario Carpentieri Department of Electrical and Information Engineering, Polytechnic University of Bari, 70125 Bari, Italy; [email protected] * Correspondence: [email protected]; Tel.: +39-080-5963-590; Fax: +39-080-5963-410 Received: 6 December 2017; Accepted: 20 December 2017; Published: 23 December 2017 Abstract: The aim of this paper is the monitoring of the energy performance of Photovoltaic (PV) plants in order to detect the presence of low-intensity anomalies, before they become failures or faults. The approach is based on several statistical tools, which are applied iteratively as the data are acquired. At every loop, new data are added to the previous ones, and a proposed procedure is applied to the new dataset, therefore the analysis is carried out on cumulative data. In this way, it is possible to track some specific parameters and to monitor that identical arrays in the same operating conditions produce the same energy. The procedure is based on parametric (ANOVA) and non-parametric tests, and results effective in locating anomalies. Three cumulative case studies, based on a real operating PV plant, are analyzed. Keywords: ANOVA; non-parametric test; unimodality; homoscedasticity; kurtosis; skewness 1. Introduction A crucial problem for the management of PV plants is the strong dependence of the system response on many extrinsic factors, such as irradiance intensity, cloudiness and pollution, humidity, air velocity, environment temperature, and cell temperature. Several models that are able to evaluate the effects of such uncertainties have been proposed in [1–9], whereas the statistical methods have been used for the design of PV plants [10,11] and for the suitable choice of the electrical components [12,13]. When a PV plant is operating, a monitoring system to check the performance in all the environmental conditions is needed. PV modules are the main components of a PV plant, so a deep attention has to be focused on their health state. For this aim, techniques that are commonly used to verify the presence of typical defects in PV modules are based on the infrared analysis [14,15], on the luminescence imaging [16], on their combination [17], while automatic procedures to extract information by the thermo-grams are proposed in [18,19]. Nevertheless, these approaches regard single modules of PV plants and they are useful when a defect has been roughly individualized. When there is no information about the general operation of the PV plant, other techniques have to be considered, as neural networks [20], check of the electrical variables [21], AutoRegressive Integrated Moving Average (ARIMA) models [22], statistics, and so on. For example, in [23], the authors proposed a decision algorithm based on both the descriptive and inferential statistics. The former one is used to characterize the data-set under investigation, on the basis of a descriptive model or a distribution family. The latter one is used for unknown dataset and consists of a data producing process with the aim to infer the characteristics of the population on the basis of a sub-set of the sample data, in order to predict anomalies, even though the amount of field measurements is limited. Nevertheless, when important failures/anomalies (short circuit, islanding, and so on) occur, the electrical variables and the associated energy have fast and not negligible variations, so they are instantaneously detected. These events can be classified as Energies 2018, 11, 30; doi:10.3390/en11010030 www.mdpi.com/journal/energies Energies 2018, 11, 30 2 of 12 high-intensity anomalies, because they produce a drastic change. Instead, the low-intensity anomalies (ageing of the components, minimal partial shading, corrosion and so on) produce a minimal variation on the values of the electrical variables and produced energy, so it is not trivial to detect them. Moreover, these anomalies can evolve in failures or faults, so a fast and correct identification can prevent them and limit the out of order. This paper proposes a cheap, easy, and fast statistical based on an algorithm that processes the data usually stored in the datalogger (Easy, Fronius, Bussolengo (VR), Italy) of the PV plants, therefore not requiring new hardware/components. The proposed methodology evaluates a priori the presence of high-intensity anomaly in each array of the PV plant, then discriminates the statistical distributions between unimodal and multimodal, and finally applies several statistical tools to extract definitive conclusions about the similarity among the energy produced by the arrays, in order to detect and locate possible anomalies. The paper presents the methodology in Section2, describes the characteristics of a real PV plant under investigation in Section3, and discusses the results in Section4. Conclusions end the paper. 2. Statistics-Based Procedure The random variability of the environmental conditions affects the irradiance intensity for the PV generators. During clear days an analytic expression for solar irradiance can be defined, whereas it is not possible for cloudy days, when, instead, the statistical approach results are very effective. In this paper, we consider that the PV plant is composed of k identical arrays with a devoted unit of measurement for each array. Each unit stores measurements of voltage, current and energy produced, whereas a central unit acquires the data coming from all the arrays. For our aims, let us consider the dataset constituted by the energy values. The descriptive statistical parameters (mean, median, and variance) are valuable tools to detect large failures (short circuits, open circuits, large shading, etc.) that provoke not negligible energy reductions. Instead, when a low-intensity anomaly appears in an array (light ageing, partial shading, and so on), the energy reduction is limited, and the previous parameters are not able to distinguish an anomaly from an acceptable tolerance. For this aim, here, we propose the methodology reported in Figure1, based on the comparison among the energy performances of the arrays. This goal can be pursued by means of parametric or non-parametric tests. When the data distribution is not normal, a non-parametric test must be used. As the non-parametric tests make only mild assumptions, they are less powerful than the parametric ones for normally distributed data. So, it is preferable to use a parametric test each time that it is possible. Among the parametric tests, the Analysis of Variance (ANOVA) [24] is an effective tool for inferential purposes and returns reliable feedback, when comparing the variance inside each array distribution and among the arrays’ distributions. In other words, ANOVA evaluates whether the differences of the mean values of the different groups are statistically significant or not. So, the null hypothesis H0 is that the means mi are equal, i.e., H0 : m1 = m2 = m3 = ··· = mk (1) versus the alternative one that at least one distribution has mean different from the others. Nevertheless, it can be applied only under three assumptions: (a) equal variance for all the distributions; (b) all the distributions are gaussian; and, (c) all of the observations are independent each other. In our case, the hypothesis (c) is always verified, because the k local measurement units are different; moreover, a modest violation for the assumptions (a) and (b) are allowed, because ANOVA is a robust test. So, before applying ANOVA test, several verifications are needed. The first one regards the unimodality of the statistical distributions; in fact, a multimodality distribution (Figure2) is surely not Energies 2018, 11, 30 3 of 12 normally distributed, thus violates the condition (b). The check on the unimodality can be done by means of skewness and kurtosis, as will be explained later. The skewness of a distribution is defined as: E(x − m)3 s = (2) k s3 being m the mean of the data x, s the standard deviation of x, and E(z) the expected value of the quantity z. From a mathematical point of view, the skewness is a third standardized moment and it measures the asymmetry of the data around the mean. For • sk = 0 the distribution is gaussian; • sk > 0 the data are spread out more to the right of the mean than to the left; and, • sk < 0 the data are spread out more to the left. Pay attention that sk = 0 is a necessary but not sufficient condition for the symmetry; in fact, symmetric distributions with sk 6= 0 cannot exist, but asymmetric distributions with sk = 0 can. The kurtosis, instead, is a measure of how outlier-prone a distribution is. Since the kurtosis can have several formulas, here, we consider the Pearson’s kurtosis less 3, i.e., E(x − m)4 ku = − 3 (3) s4 From a mathematical point of view, the kurtosis is a fourth standardized moment and results that: • ku = 0 the distribution is gaussian; • ku < 0 the distribution is less outlier-prone than the gaussian one; and, • ku > 0 the distribution is more outlier-prone than the gaussian one. Now, it results that skewness and kurtosis of a unimodal distribution have to satisfy the following constraint [25]: 6 U = s2 − k ≤ (4) k u 5 whether the distribution mode is equal to the distribution mean, and kurtosis is calculated as in (3). Instead, a more inclusive constraint, valid whether the distribution mode is different from the distribution mean, is [26]: 186 U∗ = s2 − k ≤ (5) k u 125 So, after the calculation of skewness and kurtosis, and of the mode and mean values of each distribution, if the constraint (4) (when mode = mean) or (5) (when mode 6= mean) is not satisfied, it needs to apply the Kruskal-Wallis test (K-W) [27,28] or Mood’s median test (MM), i.e., a non-parametric test, which requires only that the measurements come from a continuous distribution, even though it is not gaussian.